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INTEGRAL MONODROMY GROUPS OF KLOOSTERMAN SHEAVES

Part of: Exponential sums and character sums Families, fibrations Linear algebraic groups and related topics

Published online by Cambridge University Press:  08 June 2018

Corentin Perret-Gentil
Corentin Perret-Gentil*
Affiliation:
Department of Mathematics, ETH Zürich, Switzerland
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Abstract

We show that integral monodromy groups of Kloosterman $\ell$-adic sheaves of rank $n\geqslant 2$ on $\mathbb{G}_{m}/\mathbb{F}_{q}$ are as large as possible when the characteristic $\ell$ is large enough, depending only on the rank. This variant of Katz’s results over $\mathbb{C}$ was known by works of Gabber, Larsen, Nori and Hall under restrictions such as $\ell$ large enough depending on $\operatorname{char}(\mathbb{F}_{q})$ with an ineffective constant, which is unsuitable for applications. We use the theory of finite groups of Lie type to extend Katz’s ideas, in particular the classification of maximal subgroups of Aschbacher and Kleidman–Liebeck. These results will apply to study hyper-Kloosterman sums and their reductions in forthcoming work.

MSC classification

Primary: 11L05: Gauss and Kloosterman sums; generalizations 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.) 20G40: Linear algebraic groups over finite fields
Type
Research Article
Information
Mathematika , Volume 64 , Issue 3 , 2018 , pp. 652 - 678
Copyright
Copyright © University College London 2018 

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Footnotes

1

Current address: Centre de recherches mathématiques, Université de Montréal, Case postale 6128, Montréal QC H3C 3J7, Canada corentin.perretgentil@gmail.com

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